#124028

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,4)(6,8)(9,10,13,17,14,18,26,33)(11,19,22,34,21,15,27,29)(12,16,24,31,25,32,37,40)(20,30,35,39,23,28,36,38);(2,4)(9,11)(10,15,18,19)(12,23,25,20)(13,22)(14,21)(16,30)(17,29,33,34)(24,36,37,35)(26,27)(28,32)(31,39)(38,40);(5,6,7,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,31)(18,32)(19,30)(21,23)(22,35)(26,37)(27,36)(29,38)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,g^2=i,b^a=bd,c^a=ce,h^a=hi,c^b=cf,d^b=di,f^b=fi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31224);